Question 1

Which of the following best describes a first-order linear differential equation?

Accepted Answer

An equation of the form $$\frac{dy}{dt} + P(t)y = Q(t)$$

Question 2

Consider the differential equation $$\frac{dy}{dt} = y(y-3). $$Which of the following statements is true about its equilibrium solutions?

Accepted Answer

$$y=0$$ and $$y=3$$ are equilibrium solutions.

Question 3

Given the initial value problem $$\frac{dy}{dt} = 2y$$, $$y(0) = 5$$, what is the general solution?

Accepted Answer

$$y(t) = 5e$$^{2t}$$

Question 4

Suppose you use Euler's Method with step size $$h=0.1 to $$approximate the solution to $$y' = y$$, $$y(0) = 1 at$$ $$t=0.2. $$What is the approximate value of $$y(0.2)$$?

Accepted Answer

$$1.21$$

Question 5

Which of the following sequences is bounded above and below but does NOT converge?

Accepted Answer

$$a_n = (-1)^n$$

Question 6

A student borrows $$\50$$,000$$ at an interest rate of 0.5$$\%$$ per month and makes monthly payments of $$\$$500. $$Which differential equation models the loan balance $$L(t)$$ over time $$t (in $$months)?

Accepted Answer

$$\frac{dL}{dt} = 0.005L - 500$$

Question 7

Find the general solution to $$\frac{dy}{dx} = \frac{3y}{2x+1}.$$

Accepted Answer

$$y = C(2x+1$$)^{3/2}$$

Question 8

Which of the following best describes a sequence that is increasing and converges?

Accepted Answer

$$a_n = 1 - \frac{1}{n}$$

Question 9

Evaluate the telescoping series $$\sum_{k=2}^{\infty} \left( \frac{1}{k-1} - \frac{1}{k+1} \right).$$

Accepted Answer

$$\frac{3}{2}$$

Question 10

Which of the following is the explicit formula for the sequence $$2, 5, 8, 11, \ldots$$?

Accepted Answer

$$a_n = 3n - 1$$

Question 11

Suppose $$y' = y(y-3). If$$ $$y(0) = 1$$, what is the long-term behavior of the solution as $$t 	o \infty$$?

Accepted Answer

$$y(t)$$ approaches $$0$$

Question 12

Which of the following is a recursive formula for the sequence $$2, 5, 8, 11, \ldots$$?

Accepted Answer

$$a_{n+1} = a_n + 3$$, $$a_1 = 2$

Question 13

Which of the following best describes a sequence that is nonincreasing and diverges?

Accepted Answer

$$a_n = -n$$

Question 14

Given the differential equation $$\frac{dy}{dx} = x e$$^{x^2}$$ - \ln(y^2$)$$, which method is most appropriate to find the general solution?

Accepted Answer

Separation of variables

Question 15

A tank contains 5000 Garlic Mustard Weed plants. New plants grow at 3% per month, and 100 plants are harvested each month. Which differential equation models the number of plants $$P(t)$$?

Accepted Answer

$$\frac{dP}{dt} = 0.03P - 100$$

Question 16

Which of the following statements about the convergence of the geometric series $$\sum_{n=0}^{\infty} ar^n is $$correct?

Accepted Answer

It converges if $$|r| \u003c 1$$

Question 17

Given the learning curve model $$\frac{dP}{dt} = k(M - P)$$, where $$k \u003e 0$$, what is the general solution for $$P(t)$$?

Accepted Answer

$$P(t) = M - (M - P_0$)e^{-kt}$

Question 18

Which of the following sequences is monotonic and bounded above?

Accepted Answer

$$a_n = 1 - \frac{1}{n}$$

Question 19

Find the limit $$\lim_{n 	o \infty} e$$^{-n}$$ 2 \sin(e$$^{-n}$$).$$

Accepted Answer

$$0$$

Differential Equations and Sequences & Series: Study Guide

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